Theorem cleljust 2153

Description: When the class variables in Definition df-clel 2835 are replaced with setvar variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the setvar variables in wel 2145 with the class variables in wcel 2144. (Contributed by NM, 28-Jan-2004.) Revised to use equsexvw 2037 in order to remove dependencies on ax-10 2177, ax-12 2212, ax-13 2401. Note that there is no disjoint variable condition on 𝑥, 𝑦, that is, on the variables of the left-hand side, as should be the case for definitions. (Revised by BJ, 29-Dec-2020.)

Assertion

Ref	Expression
cleljust	⊢ (𝑥 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦))

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem cleljust

Step	Hyp	Ref	Expression
1		elequ1 2151	. . 3 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
2	1	equsexvw 2037	. 2 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦) ↔ 𝑥 ∈ 𝑦)
3	2	bicomi 227	1 ⊢ (𝑥 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812

This theorem is used by:  dfclel  2836  wl-dfclel.basic  38529